Description and implementation of an algebraic multigrid preconditioner for H1-conforming finite element schemes

Keywords: Finite element methods, H1-conforming schemes, low-order approximations, multilevel techniques, computational implementation, MATLAB®

Abstract

This paper presents detailed aspects regarding the implementation of the Finite Element Method (FEM) to solve a Poisson’s equation with homogeneous boundary conditions. The aim of this paper is to clarify details of this implementation, such as the construction of algorithms, implementation of numerical experiments, and their results. For such purpose, the continuous problem is described, and a classical FEM approach is used to solve it. In addition, a multilevel technique is implemented for an efficient resolution of the corresponding linear system, describing and including some diagrams to explain the process and presenting the implementation codes in MATLAB®. Finally, codes are validated using several numerical experiments. Results show an adequate behavior of the preconditioner since the number of iterations of the PCG method does not increase, even when the mesh size is reduced.

References

Beck, R. (1999). Graph-based algebraic multigrid for Lagrange-type finite elements on simplicial meshes. Berlin, Germany: Konrad Zuse Zentrum für Informationstechnik. Recuperado de https://pdfs.semanticscholar.org/c719/beb475ff33cab3f235c361f7e8d76e9f820c.pdf
Beirão da Veiga, L.; Brezzi, F.; Cangiani, A.; Manzini, G.; Marini, L. D. & Russo, A. (2013). Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences, 23(01), 199-214. doi: https://doi.org/10.1142/S0218202512500492
Burden, R. L.; Faires, J. D. & Burden, A. M. (2015). Numerical Analysis (10th ed.). United States: Cengage Learning.
Carstensen, C. & Klose, R. (2002). Elastoviscoplastic finite element analysis in 100 lines of Matlab. Journal of Numerical Mathematics, 10(3), 157-192. doi: https://doi.org/10.1515/JNMA.2002.157
Castillo, P. E. & Sequeira, F. A. (2013). Computational aspects of the Local Discontinuous Galerkin method on unstructured grids in three dimensions. Mathematical and Computer Modelling, 57(9-10), 2279-2288. doi: https://doi.org/10.1016/j.mcm.2011.07.032
Chen, K. (2005). Matrix preconditioning techniques and applications. United Kingdom: Cambridge University Press. doi: https://doi.org/10.1017/CBO9780511543258
Ciarlet, P. G. (2002). The finite element method for elliptic problems. United States: SIAM. doi: https://doi.org/10.1137/1.9780898719208
Cockburn, B. & Shu, C. W. (1998). The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Analysis, 35(6), 2440-2463. doi: https://doi.org/10.1137/S0036142997316712
Johnson, C. (2009). Numerical solution of partial differential equations by the finite element method. United States: Courier Corporation.
Ruge, J. W. & Stüben, K. (1987). Algebraic multigrid. In: S.F., McCormick. (Ed.), Multigrid Methods (pp. 73-130). United States: SIAM. doi: https://doi.org/10.1137/1.9781611971057.ch4
Saad, Y. (1996). ILUM: a multi-elimination ILU preconditioner for general sparse matrices. SIAM Journal on Scientific Computing, 17(4), 830-847. doi: https://doi.org/10.1137/0917054
Saad, Y. (2003). Iterative methods for sparse linear systems (2th ed.). United States: SIAM. Recuperado de https://www-users.cs.umn.edu/~saad/IterMethBook_2ndEd.pdf
Shewchuk, J. R. (1996). Triangle: engineering a 2D quality mesh generator and Delaunay triangulator. In Workshop on Applied Computational Geometry (pp. 203-222). Heidelberg, Germany: Springer. doi: https://doi.org/10.1007/BFb0014497
Stüben, K. (2001). A review of algebraic multigrid. In: C., Brezinski, L., Wuytack. (Eds.), Numerical Analysis: historical Developments in the 20th Century (pp. 331-359). Elsevier. doi: https://doi.org/10.1016/B978-0-444-50617-7.50015-X
Trottenberg, U.; Oosterlee, C. W. and Schuller, A. (2000). Multigrid. United States: Academic Press.
Wilkinson, J. H. (1994). Rounding errors in algebraic processes. United States: Courier Corporation.
Published
2020-07-31
How to Cite
Guillén-Oviedo, H., Ramírez-Jiménez, J., Segura-Ugalde, E., & Sequeira-Chavarría, F. (2020). Description and implementation of an algebraic multigrid preconditioner for H1-conforming finite element schemes. Uniciencia, 34(2), 55-81. https://doi.org/10.15359/ru.34-2.4
Section
Original scientific papers (evaluated by academic peers)